Regularizing Effect for a Nonlocal Maxwell-Schrödinger System

In this paper we prove existence and regularity of weak solutions for the following system \begin{align*} \begin{cases} &-\mbox{div}\Bigg(\bigg(|\nabla u|^{p}{L^{p}}+|\nabla v|^{p}{L^{p}}\bigg)|\nabla u|^{p-2}\nabla u\Bigg) + g(x,u,v)=f \ \ \ \mbox{in} \ Ω; &-\mbox{div}\Bigg(\bigg(|\nabla u|^{p}{L^{p}}+|\nabla v|^{p}{L^{p}}\bigg)|\nabla v|^{p-2}\nabla v\Bigg) = h(x,u,v) \ \ \ \ \mbox{in} \ Ω; &u=v=0 \ \mbox{on} \ \partialΩ. \end{cases} \end{align*} where $\mathrm{\Omega }$ is an open bounded subset of ${\mathbb{R}}^N$, $N>2$, $f\in L^m(\mathrm{\Omega })$, where $m>1$ and $g$, $h$ are two Carathéodory functions, which may be non monotone. We prove that under appropriate conditions on $g$ and $h$, there is gain of Sobolev and Lebesgue regularity for the solutions of this system.